Temperature of nitrogen in a vessel of volume $$2{m^3}$$ is $$288$$ $$K.$$ $$A$$ $$U$$-tube manometer connected to the vessel shows a reading of $$70$$ $$cm$$ of mercury (level higher in the end open to atmosphere). The universal gas constant is $$8314$$ $$J/kmol$$-$$K,$$ atmospheric pressure is $$1.01325$$ bar, acceleration due to gravity is $$9.81$$ $$m/{s^2}$$ and density of mercury is $$13600$$ $$kg/{m^3}.$$ The mass of nitrogen (in $$kg$$) in the vessel is _________

A well insulated rigid container of volume $$1{m^3}$$ contains $$1.0$$ $$kg$$ of an ideal gas $$\left[ {{C_p} =1000\,\,\,J/\left( {kg.K} \right)} \right.$$ and $$\left. {{C_v} = 800J/\left( {kg.K} \right)} \right]$$ at a pressure of $${10^5}\,\,Pa.$$ A stirrer is rotated at constant $$rpm$$ in the container for $$1000$$ rotations and the applied torque is $$100$$ $$N$$-$$m.$$ The final temperature of the gas (in $$K$$) is _______________.

A Carnot engine $$(CE$$-$$1)$$ works between two temperature reservoirs $$A$$ and $$B,$$ where $${T_A} = 900K$$ and $${T_B} = 500K.$$ A second Carnot engine $$(CE$$-$$2)$$ works between temperature reservoirs $$B$$ and $$C,$$ where $${T_c} = 300\,\,K.$$ In each cycle of $$CE$$-$$1$$ and $$CE$$-$$2,$$ all the heat rejected by $$CE$$-$$1$$ to reservoir $$B$$ is used by $$CE$$-$$2.$$ For one cycle operation, if the net $$Q$$ absorbed by $$CE$$-$$1$$ from reservoir $$A$$ is $$150$$ $$MJ,$$ the net heat rejected to reservoir $$C$$ by $$CE$$-$$2$$ (in $$MJ$$) is ______________

Steam enters a well insulated turbine and expands isentropically throughout. At an intermediate pressure, $$20$$ percent of the mass is extracted for process heating and the remaining steam expands isentropically to $$9$$ $$kPa.$$ Inlet to turbine $$P=14$$ $$MPa,$$ $$T = {560^ \circ }C,$$ $$h=3486$$ $$kJ/kg,$$ $$s=6.6$$ $$kJ/(kg.K)$$

Intermediate stage: $$h=2776$$ $$kJ/kg$$
Exit of turbine : $$P=9kPa,$$ $${h_f} = 174\,\,kJ/kg,$$
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{h_g} = 2574\,\,kJ/kg,\,\,\,{s_f} = 0.6kJ/\left( {kg.K} \right); \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,{s_g} = 8.1\,\,kJ/(kg.K) \cr} $$

If the flow rate of steam entering the turbine is $$100$$ $$kg/s,$$ then the work output (in $$MW$$) is __________.